Finite Group Databasebeta

← #36365158

Subgroups — #36365158

Order 22144.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C212yesmaximal, complemented
2#26012145C211yesmaximal, characteristic, complemented
3#26012126C211yesmaximal, characteristic, complemented
4#26011863C211yesmaximal, characteristic, complemented
5#26011736C212yesmaximal, complemented
6#26011697C211yesmaximal, characteristic, complemented
7#26011643C212yesmaximal, complemented
8#26011633C212yesmaximal, complemented
9C4×#14583811C211yesmaximal, characteristic, complemented
10#26011568C211yesmaximal, characteristic, complemented
11#459×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2212yescomplemented
13C2×D4×D173—21no—
14C2×D4×D173C2212yes—
15#19149703—28no—
16C2×#14583815C2212yes—
17C2×#14583814C2212yescomplemented
18C2×#14583814C2212yescomplemented
19C2×#14583811C2211yescharacteristic
20C2×#14583809C2212yescomplemented
21#19149679C2211yescharacteristic, complemented
22#19149663—22no—
23#19149659C2212yescomplemented
24#19149653—21no—
25#19149650—21no—
26#19149650C2212yes—
27#19149647—22no—
28C4×D692C2211yescharacteristic
29C4×D692C2211yescharacteristic
30C2×C4⋊Dic173—21no—
31#19149612C2211yescharacteristic, complemented
32#19149604C2212yescomplemented
33#19149600C2211yescharacteristic, complemented
34#19149599C2211yescharacteristic
35#19149591C2212yescomplemented
36C4×#13063436C2211yescharacteristic, complemented
37C2×D8×C173C2212yescomplemented
38#127×C173C2211yescharacteristic
39C4×D4×C173C2211yescharacteristic, complemented
40C4⋊C8×C173C2211yescharacteristic, complemented
41#102×C173C2212yescomplemented
42C23×D173—22no—
43#14583825—24no—
44#14583825—42no—
45C2×Dic346C2312yes—
46C2×D692D411yescharacteristic
47C2×D692C2311yescharacteristic
48C2×C22⋊D173—22no—
49C2×C22⋊D173—22no—
50C2×C22⋊D173—22no—
51D4×D173—24no—
52D4×D173—42no—
53D4×D173—44no—
54C22×Dic173—22no—
55C2×C4×D173—21no—
56C2×C4×D173C2312yes—
57#14583815—24no—
58#14583814—24no—
59#14583814—42no—
60#14583811D414yescomplemented
61#14583809—24no—
62C2×#13063436C2311yescharacteristic
63C2×#13063436C2311yescharacteristic
64C4⋊Dic173—22no—
65C4⋊Dic173—21no—
66C4⋊Dic173—21no—
67C4⋊Dic173D411yescharacteristic
68C4⋊Dic173C2311yescharacteristic
69C22⋊Dic173—22no—
70#14583803—22no—
71#14583803—21no—
72#14583803D412yes—
73C2×D4×C173—21no—
74C2×D4×C173—21no—
75C2×D4×C173C2312yes—
76C2×D4×C173C2311yescharacteristic
77C22×C4×C173—21no—
78D8×C173—24no—
79C2×C8×C173C2312yes—
80C4⋊C4×C173C2311yescharacteristic
81C22⋊C4×C173—21no—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—44no—
85C22×D173—44no—
86C22×D173—82no—
87C22×D173C2×D412yes—
88Dic346—22no—
89Dic346—22no—
90D692—22no—
91D692—21no—
92D692C2×D412yes—
93C22⋊D173—44no—
94C22⋊D173—44no—
95C22⋊D173—44no—
96C2×Dic173—21no—
97C2×Dic173—22no—
98C2×Dic173—44no—
99C2×Dic173C2×D412yes—
100C4×D173—22no—
101C4×D173—22no—
102C4×D173—42no—
103#13063436—21no—
104#13063436C2×D412yescomplemented
105C23×C173—21no—
106C23×C173—22no—
107D4×C173—22no—
108D4×C173—21no—
109D4×C173—22no—
110D4×C173—42no—
111D4×C173—81no—
112D4×C173C2×D412yescomplemented
113C2×C4×C173—21no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2411yescharacteristic
118C8×C173—22no—
119C22×C173—22no—
120C22×C173—41no—
121C22×C173—41no—
122C22×C173—42no—
123C22×C173—42no—
124C22×C173—42no—
125C22×C173C22×D411yescharacteristic
126C2×D173—22no—
127C2×D173—22no—
128C2×D173—22no—
129C2×D173—22no—
130C2×D173—41no—
131C2×D173—41no—
132C2×D173—88no—
133C4×C173—21no—
134C4×C173—41no—
135C4×C173AGL(1,ℤ/8)12yescomplemented
136C4×C173C22×D411yescharacteristic
137C4×C173H(5,𝔽2)11yescharacteristic
138Dic173—22no—
139Dic173—42no—
140Dic173—41no—
141C2×C173—42no—
142C2×C173—82no—
143C2×C173D4211yescharacteristic
144C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C173#57711yescharacteristic
146D173—42no—
147D173—42no—
148C173#327211yescharacteristic, complemented, Sylow
149#3272—1731nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1732no—
151#547—1731no—
152D42—1731no—
153#492—1731no—
154#468—1732no—
155#461—1731no—
156#459D17311yescharacteristic, complemented
157#450—1732no—
158#449—1732no—
159C4×D8—1731no—
160#436—1731no—
161C2×#41—1732no—
162C22×D4—3461no—
163C22×D4—1732no—
164AGL(1,ℤ/8)—3468no—
165C2×SD16—1732no—
166C2×SD16—1732no—
167C2×SD16—1732no—
168C2×D8—1731no—
169C2×D8C2×D17312yescomplemented
170C2×M16—1732no—
171C4⋊Q8—1731no—
172#127C2×D17311yescharacteristic
173#123—3462no—
174C22⋊Q8—1732no—
175#121—3461no—
176#121—3461no—
177#121—1732no—
178#120—3462no—
179C4×D4—1731no—
180C4×D4—1731no—
181C4×D4C2×D17311yescharacteristic, complemented
182C2×C4⋊C4—3461no—
183#107—1731no—
184C4⋊C8C2×D17311yescharacteristic, complemented
185#103—1732no—
186#102—1731no—
187#102—1731no—
188#102C2×D17312yescomplemented
189C22⋊C8—1732no—
190C4×C8—1731no—
191C24—3462no—
192#41—3464no—
193#41—6922no—
194C2×Q8—1732no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—3464no—
198C2×D4—3462no—
199C2×D4—3462no—
200C2×D4—3462no—
201C2×D4—1731no—
202C2×D4—1731no—
203C2×D4—6922no—
204C2×D4—6924no—
205C2×D4C22×D17312yes—
206C2×D4C22×D17311yescharacteristic
207C22×C4—21no—
208C22×C4—3462no—
209C22×C4—3461no—
210C22×C4—1732no—
211SD16—3464no—
212SD16—3464no—
213SD16—6922no—
214D8—24no—
215D8—1734no—
216M16—3464no—
217C2×C8—1731no—
218C2×C8—1731no—
219C2×C8C22×D17312yes—
220C4⋊C4—3462no—
221C4⋊C4—3461no—
222C4⋊C4—3461no—
223C4⋊C4—1731no—
224C4⋊C4—1731no—
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3462no—
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C22⋊C4—1732no—
231C42C22×D17311yescharacteristic
232C23—21no—
233C23—22no—
234C23—3461no—
235C23—13842no—
236C23—1732no—
237C23—6924no—
238C23—6924no—
239Q8—3462no—
240Q8—3462no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—81no—
246D4—3462no—
247D4—3461no—
248D4—1732no—
249D4—6924no—
250D4—6924no—
251D4—6924no—
252D4D4×D17312yescomplemented
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—3462no—
256C2×C4—3462no—
257C2×C4—3461no—
258C2×C4—3462no—
259C2×C4—1732no—
260C2×C4—6924no—
261C2×C4—6922no—
262C2×C4D4×D17311yescharacteristic
263C2×C4D4×D17311yescharacteristic
264C2×C4C23×D17311yescharacteristic
265C8—22no—
266C8—3461no—
267C8—1732no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—3462no—
275C22—3462no—
276C22—3462no—
277C22—3462no—
278C22—13848no—
279C22—6921no—
280C22—6921no—
281C22C2×D4×D17311yescharacteristic
282C4—21no—
283C4—41no—
284C4—3462no—
285C4—6921no—
286C4—6922no—
287C4#1914970312yescomplemented
288C4C2×D4×D17311yescharacteristic
289C4#1914972511yescharacteristic
290C2—42no—
291C2—82no—
292C2—6922no—
293C2—6922no—
294C2#2601212511yescharacteristic
295C2C2×#1914970311yescharacteristic
296C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. #26012145C2semidirect
3. #26012126C2semidirect
4. #26011863C2semidirect
5. #26011736C2semidirect
6. #26011697C2semidirect
7. #26011643C2semidirect
8. #26011633C2semidirect
9. C4×#14583811C2semidirect
10. #26011568C2semidirect
11. #459×C173C2semidirect
12. C2×#14583825C22semidirect
17. C2×#14583814C22semidirect
18. C2×#14583814C22semidirect
20. C2×#14583809C22semidirect
21. #19149679C22semidirect
23. #19149659C22semidirect
31. #19149612C22semidirect
32. #19149604C22semidirect
33. #19149600C22semidirect
35. #19149591C22semidirect
36. C4×#13063436C22semidirect
37. C2×D8×C173C22semidirect
39. C4×D4×C173C22semidirect
40. C4⋊C8×C173C22semidirect
41. #102×C173C22semidirect
60. #14583811D4semidirect
104. #13063436C2×D4semidirect
112. D4×C173C2×D4semidirect
135. C4×C173AGL(1,ℤ/8)semidirect
148. C173#3272semidirect
156. #459D173semidirect
169. C2×D8C2×D173semidirect
181. C4×D4C2×D173semidirect
184. C4⋊C8C2×D173semidirect
188. #102C2×D173semidirect
252. D4D4×D173semidirect
287. C4#19149703semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191497032
#260121452
#260121262
#260118632
#260117362
#260116972
#260116432
#260116332
C4×#145838112
#260115682
#459×C1732
#3272173